Beauville proves that \(\ker (\alpha )\)  is a finite group of order \(2^{k-1}\)  (see [1]}) which implies that there is no nontrivial group homomorphism \(\mathbb {C}^* \rightarrow \ker (\alpha )\) . The snake lemma then implies that there is a short exact sequence
\(1 \rightarrow (\mathbb {C}^*)^k \rightarrow \ker (1-\sigma ) \rightarrow \widetilde{\phi }^*\textup {Pic}^0(\widetilde{D}) \times (\mathbb {Z}/2\mathbb {Z})^{k-1} \rightarrow 1,\) 
